through π:UCof(BA⊗A)→BA⊗A\pi: U Cof(B^{A \otimes A}) \to B^{A \otimes A}. Here δ\delta denotes the comultiplication (same as the diagonal map as seen in CoalgCoalg), and ⊗1\otimes_1 indicates the structure of enriched functoriality for ⊗\otimes.

The map Ψ:Cof(BA)→Cof(Bk)\Psi: Cof(B^A) \to Cof(B^k) is the unique coalgebra map such that UΨU \Psi lifts the map